2 Answers
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In campaign, you build a team and start your way through guiding that team towards winning better and better trounamets. Main point of the campaign is to develop the team gradually. This can be quite a lenghty endevor, since there are quite a few tournaments to get through, each tournamet takes several matches and there's a very real risk of not being able to advance to better tournaments if you don't do well enough. You'll also be able to see your team and its player grow and learn as they gather experience, get injured and sometimes even die.

In story mode, you play as a coach that has a very bad habbit of getting kidnapped to a new team over and over again. You'll rarely spend more than a match or two with a team (sometimes even less as you may be given just one half-time) and your goals vary significantly. In stead of aiming for just the victory, you'll be given goals like performing certain amount of passes or fouls in a match, getting certain amoung of touchdowns and such. Some of these task can be quite difficult to pull. You'll also get some humorous backstory in form of sports news between matches and after failures, where the news detail just how you met your demise at the hands of disappointed players/fans/managment.

The campaign is a management game, in which you create your own team and lead it trying to reach the top of the league.
In story mode instead you are a mercenary trainer, playing with different teams and following a story "that will guide you to Blood Bowl’s origins".

what do you mean by mercenary trainer though? I read that and I still don't understand. What exactly does a trainer do? As near as I can tell both modes end up with you controlling a team in a match.
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z 'Apr 19 '12 at 12:47

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In story mode you're hired as assistant coach by a series of team. Each head coach wil give you directions to perform various actions on the field, and eventually, going on, you'll become coach yourself. I think they added the story mode to let the player gain familiarity with the various teams and their different characteristics.
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KappeiApr 19 '12 at 12:54